Theorem sbieh 1670

Description: Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1671 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)

Hypotheses

Ref	Expression
sbieh.1	⊢ (ψ → ∀xψ)
sbieh.2	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
sbieh	⊢ ([y / x]φ ↔ ψ)

Proof of Theorem sbieh

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (φ → φ)
2	1	hbth 1349	. . 3 ⊢ ((φ → φ) → ∀x(φ → φ))
3		sbieh.1	. . . 4 ⊢ (ψ → ∀xψ)
4	3	a1i 9	. . 3 ⊢ ((φ → φ) → (ψ → ∀xψ))
5		sbieh.2	. . . 4 ⊢ (x = y → (φ ↔ ψ))
6	5	a1i 9	. . 3 ⊢ ((φ → φ) → (x = y → (φ ↔ ψ)))
7	2, 4, 6	sbiedh 1667	. 2 ⊢ ((φ → φ) → ([y / x]φ ↔ ψ))
8	1, 7	ax-mp 7	1 ⊢ ([y / x]φ ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-sb 1643

This theorem is referenced by:  sbie  1671  sbco2vlem  1817  equsb3lem  1821  sbco2yz  1834  elsb3  1849  elsb4  1850  dvelimf  1888